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The nonlinear (NL) elastic behavior of rocks evidenced by the stress-induced variations of the seismic velocities and related to the presence of mechanical defects (e.g. cracks) is now well established. Another classical result is the anisotropic behavior of rocks due to the spatial order exhibited by the orientation distribution function of heterogeneities (e.g. aligned microfractures, preferred orientation of grains). Both linear and NL elastic properties of the rock are affected by this order. We propose a global method for simultaneously characterizing anisotropy and nonlinearity in anisotropic media of arbitrary symmetry, such as rocks, based on a vectorial mapping of the elastic tensors associated with a suitable metric. The method provides the tools to (a) quantify linear elasticity and NL elasticity; (b) compare linear anisotropy and NL anisotropy; and (c) replace the considered medium of low symmetry by a simpler medium of higher symmetry, i.e. isotropic, transversely isotropic or orthotropic. The analysis of experimental data shows that rocks exhibit strikingly strong nonlinearity, orders of magnitude larger than most “intact” homogeneous materials (i.e. media without mechanical defects, such as crystals). In all the considered materials NL anisotropy is larger than linear anisotropy, but a material exhibiting a larger NL anisotropy does not necessarily exhibit larger linear anisotropy. For instance NL anisotropy can be comparable to larger in rocks than in their constituent minerals, whereas the linear anisotropy is weaker in the former than in the latter.

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